What is the time complexity of Dijkstra's algorithm when implemented with a bina

What is the time complexity of Dijkstra's algorithm when implemented with a binary heap?

What is the time complexity of Dijkstra's algorithm when implemented with a binary heap?

Step 1: Understand that Dijkstra's algorithm is used to find the shortest path from a starting vertex to all other vertices in a graph.
Step 2: Recognize that the algorithm processes each vertex and edge in the graph.
Step 3: Identify that a binary heap is used to efficiently manage the priority of vertices based on their current shortest distance.
Step 4: Note that the algorithm involves two main operations: extracting the minimum distance vertex and updating the distances of its neighbors.
Step 5: Realize that extracting the minimum from a binary heap takes O(log V) time, where V is the number of vertices.
Step 6: Understand that for each vertex, we may need to update the distances of its adjacent edges, which can be up to E edges in total.
Step 7: Combine the operations: for each of the V vertices, we perform O(log V) for extraction and O(E) for updating edges.
Step 8: Conclude that the overall time complexity is O(E log V) because we process each edge and vertex in the graph.

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